FIGURE 3.9 Data obtained from separate experiments for the growth/no-growth (G/NG) boundary of Escherichia coli. Data are from Salter et al. (2000) (circles) and from unpublished results of the authors (diamonds). Near the G/NG boundary, the data obtained from discrete experiments do not form a smooth (monotonie) boundary, suggesting that small differences in experimental procedures can signi cantly affect the position of the boundary. Open symbols denote no-growth conditions, and solid symbols indicate that growth was observed.

The above approaches can be considered to be deterministic; i.e., they predict only one position (Pgrowth = 0.5) for the boundary, although the position of boundaries can be adjusted by "weighting" data in the case of Masana and Baranyi (2000b) or by selecting an appropriate value for 9 in the case of the Le Marc et al. (2002) approach (see Section 3.2.3). While the data of Masana and Baranyi (2000b) included tenfold replication, the midpoints of the most "extreme" conditions that did allow growth, and the least "extreme" conditions that did not allow growth were estimated by interpolation and considered to represent 50% probability of growth. Other workers have suggested that some problems require higher levels of con dence that growth will not occur, so that methods that enable de nition of the interface at selected levels of statistical con dence may have greater utility.

Another approach that implicitly characterizes the G/NG interface is that of combined growth and death models in which the rate of growth and rate of death under specified conditions are estimated simultaneously. The G/NG interface can be inferred from those combinations of conditions where growth rate and death rate are equal (see, e.g., Jones and Walker, 1993; Jones et al., 1994). A similar approach is evident in Battey et al. (2001) who modeled both the rates of growth and rates of death of spoilage molds in ready to drink beverages. The G/NG interface was given, implicitly, as that set of conditions where the rate of growth was equal to the rate of death.

Ratkowsky et al. (1991) noted that as environmental conditions become more inhibitory to microbial growth the variability in growth rates increases widely, which implies that the probability that growth occurs at all becomes uncertain, because the left-hand tail of the growth rate distribution falls below zero. This is supported in the results of Whiting and colleagues (Whiting and Call, 1993; Whiting and Oriente, 1997; Whiting and Strobaugh, 1998), where Pmax (the proportion of spores that successfully germinated and initiated growth) was shown to decline at increasingly stringent conditions. Conversely, Masana and Baranyi (2000b) observed, as have other workers (McKellar and Lu, 2001; Presser et al., 1998; Salter et al., 2000; Tienungoon et al., 2000), that the difference in conditions that allow growth and those that do not is usually abrupt, and often at or beyond the limits of resolution of instruments commonly used to measure such differences. Thus, Masana and Baranyi (2000b) questioned the need for approaches that model the transition between conditions leading to high probability of growth and those leading to low probabilities of growth. While this abrupt transition appears consistent within replicated experiments it is less certain, however, that the same consistency is true between experiments. Figure 3.9, showing experimental data, suggests that responses near the boundary may be inconsistent when data from several discrete experiments are combined. This may suggest subtle, but important, differences in response related to the physiology of the inoculum, or its concentration. Furthermore, it suggests that the ability to characterize probabilities of growth under specified sets of conditions may be an important element of growth boundary models and that the boundary may not be "absolute" but depend on the physiological state of the cell and, by inference, on the size of the inoculum. This will be discussed further in Section 3.4.4.

Ratkowsky and Ross (1995) and others (Bolton and Frank, 1999; Jenkins et al., 2000; Lanciotti et al., 2001; Lopez-Malo et al., 2000; McKellar and Lu, 2001; Parente et al., 1998; Stewart et al., 2001, 2002; Uljas et al., 2001) reintroduced the use of logistic regression to model categorical data (i.e., growth or no growth) in predictive microbiology, enabling probabilistic determination of the G/NG boundary. Use of the logit function enabled the probability of growth under specific sets of conditions to be estimated, so that the G/NG boundary could be specified at selected levels of con dence.

Ratkowsky and Ross (1995) aimed to model absolute limits to growth in mul-tifactorial space, but only had available data based on a 72-h observation period. While most other workers have preferred polynomial functions to describe the effect of independent variables on the logit function, in the former approach a square-root-type kinetic model was ln-transformed and used as the basis of the function relating the logit of probability of growth to independent variables, e.g., temperature, water activity, pH. This approach was adopted in an attempt to retain some level of biological interpretability of the models, a desire echoed by others (Augustin and Carlier, 2000a,b; Le Marc et al., 2002).

The form of the G/NG interface model of Presser et al. (1998) was derived from the kinetic model of Presser et al. (1997) for the growth rate of E. coli (see Equation

3.10). Novel data were generated specifically to assess the limits of E. coli growth under combinations of temperature, pH, aw and lactic acid. The corresponding G/NG model had the form:

+ 6.96ln[1 - LAC/(10.7 X (1 + 10ph"3 86))] + 3.06ln[1 - LAC/(823 X (1 + 103 86-ph))]

where all terms are as de ned in Section 3.2.1.

Some parameters in that model had to be determined independently, i.e., were not determined in the regression, and were derived from the fitted values of square-root-type kinetic models. Essentially the same approach was adopted by Lanciotti et al. (2001) to develop G/NG models for B. cereus, S. aureus, and Salmonella enteritidis. Ratkowsky (2002) commented on the increased exibility in being able to determine all of the parameters in the model during the regression, and subsequent studies developed the approach, eventually leading to a novel nonlinear logistic regression technique (Salter et al., 2000; Tienungoon et al., 2000). Ratkowsky (2002) pointed out that nonlinear logistic regression was a new statistical technique and discussed bene ts and problems with that approach specifically in relation to growth limits modeling. A problem with models of the form of Equation 3.44 is that for conditions more extreme than the parameters corresponding to Tmin, pHmin, aw min, etc., and which are tested experimentally though not expected to permit growth, the terms containing those parameters would become negative. As all of those terms are associated with a logarithmic transformation, the expression cannot be calculated during the regression and such values are ignored in the model tting process, or have to be eliminated from the data set before the tting process begins. This, in turn, affects the values of the parameters of the fitted model. Ratkowsky (2002) comments that an objective method for selection and deletion of such data is necessary, but does not yet exist.

Bolton and Frank (1999) extended the binary logistic regression approach by recoding growth and no growth data to allow a third category: survival, or stasis. They termed this approach ordinal logistic regression. Parente et al. (1998) "reversed" the response variable, and applied logistic regression techniques to the probability of survival/no survival of L. monocytogenes in response to bacteriocins, pH, EDTA, and NaCl. Stewart et al. (2001) modeled the probability of growth of S. aureus within 6 months of incubation at 37°C, and at reduced water activity achieved by various humectants. They also compared the growth boundary with the boundary for enterotoxin production, and observed a close correlation between the two criteria.

Growth limits models have also been developed for spoilage organisms including

Saccharomyces cerevisiae (Lopez-Malo et al., 2000) and Zygosaccharomyces bailii (Jenkins et al., 2000) and cocktails of Saccharomyces cerevisiae, Zygosaccharomyces bailii, and Candida lipolytica (Battey et al., 2002). Interestingly, the study of Jenkins et al. (2000), while encompassing broader ranges of factor combinations, con rmed the simpler and earlier model of Tuynenburg Muys (1971). That model, which specifies combinations of molar salt plus sugar and percent undissociated acetic acid for stability of acidic sauces, still forms the basis of the industry standard for those products. This observation suggests that limits to growth under combined conditions can be highly reproducible.

3.4.3.3 Relationship to the Minimum Convex Polyhedron Approach

The concept of the MCP was introduced by Baranyi et al. (1996) (see Figure 3.10) to describe the multifactor "space" that just encloses the interpolation region of a predictive kinetic model. If the interpolation region exactly matched the growth region of the organism then the MCP would also describe the growth limits of the organism. In practice, however, it would be impossible to undertake suf cient measurements to completely de ne the MCP; i.e., the MCP has "sharp" edges because of the method of its calculation, whereas from available studies (see Figures 3.8 and 3.9) the G/NG interface forms a continuously curved surface. However, it might also be possible to use no-growth data to create a no-growth MCP and to combine the growth MCP and no-growth MCP to de ne a region within which the G/NG boundary must lie. This approach has been assessed and compared to a model of the form of Equation 3.43 by Le Marc and colleagues (Le Marc et al., 2003). These workers concluded that the logistic regression modeling approach produced a smoother response surface, more consistent with observations, but that the MCP approach had the advantage of being directly linked to observations and therefore was not a prediction from a model.

FIGURE 3.10 Interpolation region (MCP) for a model that includes four-factor combinations (T, pH, NaCl, NaNO2). The interpolation region shown is that for NaCl = 0.5%, but is based on the complete data set. Solid circles indicate conditions under which observations have been made, while the lines represent the edges of the MCP. (From Masana, M.O. and Baranyi, J. Food Microbiol., 17, 367-374, 2000a. With permission.)

FIGURE 3.10 Interpolation region (MCP) for a model that includes four-factor combinations (T, pH, NaCl, NaNO2). The interpolation region shown is that for NaCl = 0.5%, but is based on the complete data set. Solid circles indicate conditions under which observations have been made, while the lines represent the edges of the MCP. (From Masana, M.O. and Baranyi, J. Food Microbiol., 17, 367-374, 2000a. With permission.)

Recently, Hajmeer and Basheer (2002, 2003a,b) demonstrated the use of a Probabilistic Neural Network (PNN) approach to de nition of the G/NG interface. PNNs are a form of ANN (see Section 3.2.6). In a series of papers, based on modeling the data of Salter et al. (2000) for the effects of temperature and water activity (due to NaCl) on the growth limits of E. coli, Hajmeer and Basheer concluded that their PNN models provided a better description of the data of Salter et al. (2000) than did the nonlinear logistic regression method referred to above. Their conclusion is considered in more detail in Section 3.4.3.5 below.

It should be noted that neither the logistic regression models described above, nor the PNN, produce an equation that describes the interface. Rather, the output of those models is the probability that a given set of conditions will allow growth. To de ne the interface, it is necessary to rearrange the model for some selected value of P to generate an equation that describes the G/NG boundary.

3.4.3.5 Evaluation of Goodness of Fit and Comparison of Models

Methods for evaluation of performance of logistic regression models include the receiver operating curve (ROC; also referred to as the concordance rate), the Hos-mer-Lemeshow goodness-of- t statistic, and the maximum rescaled R2 statistic. These are considered in greater detail in Tienungoon et al. (2000).

Brie y, the ROC is obtained from the proportion of events that were correctly predicted compared to the proportion of nonevents that were correctly predicted. The closer the value is to 1, the better the level of discrimination. In epidemiological studies, ROC values > 0.8 are considered excellent. ROC values for G/NG models are typically much higher.

The Hosmer-Lemeshow index involves grouping objects into a contingency table and calculating a Pearson chi-square statistic. Small values of the index indicate a good t of the model.

The maximum rescaled R2 value is proposed for use with binomial error as an analogy to the R2 value used with normally distributed error. The closer the value is to 1, the greater is the success of the model in predicting the observed outcome from the independent variables. Zhao et al. (2001) cite the deviance test and graphical tools such as the index plot and half normal plot as methods for determining goodness of t of linear logistic regression models.

Other methods based on calculation of indices from the "confusion matrix" (Hajmeer and Basheer, 2002, 2003b) or the equivalent "contingency matrix" (Hajmeer and Basheer, 2003a) were used to compare performance between models derived from different approaches and applied to the same data.

Another method of evaluation is to compare the fitted model to independent data sets (Bolton and Frank, 1999; Masana and Baranyi, 2000b; Tienungoon et al., 2000) although, generally, such data are not readily available (see, e.g., McKellar and Lu, 2001). The model of Tienungoon et al. (2000) for L. monocytogenes growth boundaries showed very good agreement with the data of McClure et al. (1989) and George et al. (1988) despite that different strains were involved. There is also a remarkable level of similarity between the observations of Tienungoon et al (2000) and the observations of Le Marc et al. (2002) on growth limits of L. innocua. Several publications, however, report growth of L. monocytogenes at temperatures lower than the minimum growth limit predicted by the Tienungoon et al. (2000) model, possibly indicating strain variation or that the experimental design failed to recognize important elements that facilitate L. monocytogenes growth at temperatures < 3°C, i.e., that an inappropriate growth substrate was used. Similarly, McKellar and Lu (2001) reported that their model failed to predict growth of E. coli O157:H7 under conditions where it had been previously reported, although it should be noted that their model was limited to observation of growth within 72 h. Bolton and Frank (1999) compared the predictions of their growth limits models for L. monocytogenes in cheese to the data of Genigeorgis et al. (1991) for L. monocytogenes growth in market cheese. The models predicted correctly in 65% of trials (42-day model) and 81% of trials (21-day model).

Given the diversity of approaches, it is pertinent to ask: does one method for de ning the G/NG interface perform better than another? As with kinetic models, the ability to describe a specific experimental data set does not necessarily reflect the ability to predict accurately the probability of growth under novel sets of conditions. While measures of performance of logistic regression models are available, they can be readily affected by the data set used. Perfect agreement between observed and modeled data responses may not be possible if there are anomalies in the data. Figure 3.11 provides a clear example. Nonetheless, for many growth limits models high rates of concordance (typically >90%) have been reported. As noted earlier, in epidemiological logistic regression modeling, rates higher than 70% are considered to represent good ts to the data, implying that the limits to microbial growth are highly reproducible when well-controlled experiments are conducted.

To date, only one direct comparison of G/NG modeling approaches has been presented (Hajmeer and Basheer, 2002, 2003a,b) but this was based on one data set only, i.e., that of Salter et al. (2000) for the growth limits of E. coli in temperature/water activity space. Only by comparing the performance of different modeling approaches applied to multiple data sets does an appreciation of overall model performance emerge. Nonetheless, to illustrate differences between models and give some appreciation of their overall performance we compare several models using the data of Salter et al. (2000) for the growth limits of E. coli R31 in response to temperature and water activity. The model types compared are:

1. The PNN of Hajmeer and Basheer (2003a), which those authors were able to summarize as a relatively simple equation

2. A model of the type of Equation 3.44 fitted to a subset of the Salter et al. (2000) data set by Hajmeer and Basheer (2003a) (It should be noted that, contrary to what is stated in that publication, the model presented by Hajmeer and Basheer was not generated by nonlinear logistic regression but by a two-step linear logistic regression as described in Presser et al., 1998)

Temperature (°C)

FIGURE 3.11 Comparison of predicted no growth boundaries for four modeling approaches applied to the data of Salter et al. (2000) (circles) for the growth limits of Escherichia coli R31 in response to temperature and water activity (NaCl) combinations. Approach of Hajmeer and Basheer PNN (heavy solid line); Linear Logistic /Equation 3.44 (heavy dashed line); Le Marc et al. (2002) (light solid line); Augustin and Carlier (2000a) (light dashed line). The data set was subsequently augmented with new data (diamonds), which reveals that none of the models extrapolate reliably. (Solid symbols: growth; open symbols: no growth.)

Temperature (°C)

FIGURE 3.11 Comparison of predicted no growth boundaries for four modeling approaches applied to the data of Salter et al. (2000) (circles) for the growth limits of Escherichia coli R31 in response to temperature and water activity (NaCl) combinations. Approach of Hajmeer and Basheer PNN (heavy solid line); Linear Logistic /Equation 3.44 (heavy dashed line); Le Marc et al. (2002) (light solid line); Augustin and Carlier (2000a) (light dashed line). The data set was subsequently augmented with new data (diamonds), which reveals that none of the models extrapolate reliably. (Solid symbols: growth; open symbols: no growth.)

3. A model of the type of Le Marc et al. (2002; Equation 3.25 to Equation 3.27), where Tmax = 49.23°C (to be consistent with the logistic regression model parameter), aw min = 0.948, and Tmin = 8.8°C, based on the minimum water activity and temperature, respectively, at which growth were observed

4. A model of the type of Augustin and Carlier (2000b; Equation 3.24) assuming that Tmin = 8.8°C and aw min = 0.948, consistent with the parameter values used for the Le Marc et al. (2002) model

The predicted interfaces are shown in Figure 3.11, together with the data used to generate the models. (Note that the subsets of 143 of the 179 data of Salter et al. (2000) used by Hajmeer and Basheer (2003a) to t the PNN and the Equation 3.44 type model were not identi ed.)

When compared to the full data set, the level of misprediction ranged from ~15 to 20 of the 179 data points for each of the models, suggesting that the level of performance was not greatly different despite very different modeling approaches.* A complication in the comparison of G/NG model performance is that most of the

* It should be noted that this analysis disagrees with the results of Hajmeer and Basheer (2002) who reported only two to four mispredictions for the total (i.e., training and validation) data set.

data are readily predicted, e.g., those that fall outside the known limits for growth for individual environmental factors when all other factors are optimal. Such data can "overwhelm" the data in which one is most interested, i.e., in the relatively narrow region where factors interact to reduce the biokinetic ranges and, yet more specifically, where the probability of growth rapidly changes from "growth is very likely" to "growth is very unlikely." These data de ne the interface and, consequently, data closest to the interface are more important when comparing model performance. This has implications for experimental design, as discussed in Section 3.4.4 below.

To assess whether one model might be preferred on theoretical grounds, as adjudged by its ability to extrapolate reliably, the predictions of all models in the temperature range above 35°C can be compared to data subsequently generated, shown in Figure 3.11, and not used to generate the models. Clearly, none of the models extrapolate well.

From the above comparison, it appears that despite very different modeling approaches and degrees of complexity of modeling, there is currently little to differentiate those approaches on the basis of their ability to describe the G/NG interface or on their ability to predict outside the interpolation region.

3.4.4 Experimental Methods and Design Considerations

As suggested above, currently there is little mechanistic understanding of how environmental factors interact to prevent bacterial growth and it must be recognized as a possibility that there is no single, common mechanism underlying the observed boundaries for different factor combinations. Consequently, it is not possible from rst principles to design the optimal experiment that captures the essential information that will characterize the response and lead to reliable models. Instead, at this time, experimental methods must be focused toward gaining enough data in the interface region to be able to describe empirically the limits to growth.

First of all, two approaches may be distinguished that could affect the experimental methods chosen. In one, the interest is in whether growth/toxin production, etc. is possible within some specified time limit, which may be related to the shelf life of the product. In other approaches, the objective is to de ne absolute limits to growth, i.e., the most extreme combinations of factors that just allow growth. McKellar and Lu (2001) argue that there is always a time limit imposed on G/NG modeling studies. Strategies exist, however, that provide greater con dence that the "absolute" limits to growth are being measured. Some of these are discussed below.

Several groups have assessed both growth and inactivation in their experimental treatments (McKellar and Lu, 2001; Parente et al., 1998; Presser et al., 1999; Razavilar and Genigeorgis, 1998). In this way the position of the boundary is inferred from two "directions." If growth is not observed, an observer cannot be sure whether growth is not possible or has not occurred yet. If it is known that at some more extreme condition inactivation occurs, it can be inferred that the G/NG boundary lies between those two sets of conditions.

A potential problem with this strategy is that cultures can initially display some loss of viability, but with survivors eventually initiating growth; i.e., population decline cannot unambiguously be interpreted as "growth is not possible." Numerous studies (e.g., Mellefont et al., 2003) have demonstrated that rapid transfer of a culture from one set of conditions to another that is more stressful can induce injury and death, but that survivors will eventually adjust and be able to grow. This has been termed the Phoenix phenomenon (Shoemaker and Pierson, 1976). Such regrowth has been reported in the context of G/NG modeling (Bolton and Frank, 1999; Masana and Baranyi, 2000b; Parente et al., 1998; Tienungoon et al., 2000).

Clearly, an experimenter interested in determining the "absolute" G/NG boundary will need to maximize the resistance of the inoculum to stress on exposure to the new, more stressful, environment. The use of stationary phase cultures as inocula would seem to be a minimum requirement. It may be necessary to habituate cultures to the test conditions (e.g., growth at conditions just less harsh) prior to inoculation into the test conditions to maximize the chance that growth, if possible, will be observed. One way to maximize the likelihood of observing the most extreme growth limits would be to use cultures growing at the apparent limits as inocula into slightly more stringent conditions. This also has the advantage of minimizing growth lags on inoculation into a harsher environment.

Masana and Baranyi (2000b) indicated that inoculum size affected the position of the boundary. Robinson et al. (1998) reported similar effects of inoculum density on bacterial lag times. While it is clear that time to detection would depend on inoculum density, growth detection methods were not cell-density-dependent in those studies. Parente et al. (1998) also reported that a decrease in inoculum size decreased the probability of survival. If the shock of transfer is known to inactivate a xed proportion of the cells in the inoculum, to develop a robust model it will be necessary to use an inoculum that ensures that even after inactivation there is a high probability that at least one cell will survive.

The above observations lend support to the hypothesis that it is the distribution of physiological states of readiness to survive and multiply in a new environment that determines the position of the G/NG boundary, i.e., all other things being equal, the more cells in the inoculum the more likely it is that there is one cell that has the capacity to survive and grow. This also reinforces the equivalence between G/NG boundary modeling and the modeling of conditions that lead to in nite lag times. The importance of the distribution of lag times on the observed lag times of whole populations has been discussed by Baranyi (1998).

There may be more involved reasons for inoculum density-dependent responses also, such as chemical messaging between cells (see, e.g., Miller and Bassler, 2001; Winans and Bassler, 2000).

In conclusion, if the aim is to determine absolute limits to growth, a higher number of cells is preferable. Masana and Baranyi (2000b) stated that growth boundaries "represented the chance of growth for each sample; therefore, to assure a low probability of growth in many samples, it will be more relevant to consider boundaries for high inoculum levels." Equally, as noted above, steps to maximize the cell's chances of survival and growth in the environment are also recommended.

There is potentially a caveat, however, that needs to be applied. Maximum population densities of batch cultures are reported to decline under increasingly harsh growth conditions. Thus, the use of high inocula may mask the true position of the G/NG boundary if the inoculum used is already denser than the MPD of the organism in a very stressful test environment.

3.4.4.3 Are There Absolute Limits to Microbial Growth?

In the above discussion it has been implicitly assumed that there are absolute limits to microbial growth under combined environmental stresses. It is pertinent to examine this assumption.

Numerous authors have noted that, within an experiment, the transition between conditions that allow growth, and those that do not, is abrupt and that usually all replicates at the last growth condition grow, while all the replicates at the rst-growth-preventing condition do not (Masana and Baranyi, 2000; Presser et al., 1998; Tienungoon et al., 2000). McKellar and Xu (2001), for example, reported that of 1820 conditions tested, all ve replicates of each condition either grew or did not grow. This abruptness, however, is not always evident in the modeled results (Tienungoon et al., 2000).

Conversely, between experiments by the same researcher, using the same methods and the same strain, results are not always reproducible. Figure 3.9 provides an example and Masana and Baranyi (2000b) make the same observation of their data for Brochothrix thermosphacta. At the same time, however, there is evidence of excellent reproducibility of boundaries between independent workers, using different strains, and different methods in different locations. The results of Tienungoon et al. (2000) were highly consistent between two strains tested, and more notably, with those of George et al. (1988) and Cole et al. (1990) presented a decade earlier, including different strains in one case. There is also a remarkable similarity between the pH/temperature G/NG interface of Listeria innocua reported by Le Marc et al. (2002) and the same interface for two species of L. monocytogenes presented in Tienungoon et al. (2000).

Jenkins et al. (2000) noted that the boundary they derived for the growth limits of Zygosaccharomyces baillii in beverages was very consistent with a model developed 30 years earlier for the stability of acidic sauces.

Stewart et al. (2002) noted that with S. aureus, as conditions became increasingly unfavorable for growth, the contour lines (Pgrowth) they generated drew closer and closer together, suggesting that conditions were approaching absolute limits that do not allow growth. Conversely, there are examples where one group's observations do not agree well with another's for an analogous organisms/environmental pair (e.g., Bolton and Frank, 1999; McKellar and Xu, 2001). Delignette-Muller and Rosso (2000) reported strain variability in the minimum temperature for growth.

While the above discussion points to heterogeneity in the physiological readiness of bacteria to grow in a new environment, Masana and Baranyi (2000b) also infer that differences in microenvironments, particularly within foods, could also be a source of heterogeneity in observed growth limits.

In conclusion, there is a body of experimental evidence that suggests that growth boundaries, if carefully determined, might be highly reproducible. Conversely, counterexamples exist. It remains to be determined whether the incongruous results arise from signi cant and measurable differences in methodology, e.g., the detection time used in the respective studies, or are due to uncontrollable sources (Table 3.7).

As noted above, it is not possible from rst principles to design the optimal experiment that captures the information to characterize the G/NG boundary. Various authors have suggested physiological interpretations (Battey et al, 2001; Battey and Schaffner, 2001; Jenkins et al., 2000; Lopez-Malo et al., 2000; McMeekin et al., 2000) but none have yet been experimentally tested.

Thus, an empirical approach that aims to collect as much information in the region of most interest, i.e., the G/NG interface, is recommended by most workers. Several groups of researchers have indicated that they use a two-stage modeling process. The rst uses a coarse grid of conditions of variables to roughly establish the position of the boundary. The second phase monitors responses at conditions near the boundary and at close intervals of the environmental parameters. Variable combinations far from the interface, at which growth is either highly likely or highly unlikely, do not provide much information to the modeling process, which seeks to de ne the interface with a high level of precision. Equally, it is ideal to use a design that gives roughly equal numbers of conditions where growth is, and growth is not, observed (Jenkins et al., 2000; Legan et al., 2002; Masana and Baranyi, 2000b, Uljas et al., 2001). Pragmatically, Legan et al. (2002) recommend setting up "marginal" and "no-growth" treatments rst because these treatments will run for the longest time (possibly several months). Those conditions in which growth is expected to be relatively quick can be set up last because they only need monitoring until growth is detected.

The nature of these studies necessarily involves long incubation times. Legan et al. (2002) noted that particular care must be taken to ensure that the initial conditions do not change over time solely as a result of an uncontrolled interaction with the laboratory environment. Prevention of dessication or uptake of water vapor requires particular attention. Changes resulting from microbial activity may, however, be an important part of the mechanism leading to growth initiation and should not be stabilized at the expense of growth that would naturally occur in a food. Legan et al. (2002) comment that, for example, maintaining the initial pH over time is typically neither possible nor practical, even in buffered media, and that allowing a change in pH due to growth of the organism more closely mimics what would happen in a food product than maintaining the initial pH over time.

From the above discussion, unambiguous de nition of the G/NG boundary of an organism in multidimensional space presents several paradoxical challenges. While an experimenter will do well to remember these considerations in the interpretation of his/her results, it seems probable that methods that have been used to date will have come close to identifying the "true" G/NG boundary, and that the position of the boundary will move only slightly if an experimenter acts to control all of the above variables and to maximize the potential for the observation of growth in the chosen experimental system.

While the discussion has not focused specifically on appropriate methods for probability of growth within a defined time, many of the same principles and considerations will apply.

Moreover, the field of growth limits modeling, while having an equally long history as kinetic modeling, now seems to be quite disjointed, with little rigorous comparison of approaches, let alone agreement on the most appropriate model structures or experimental methods. In particular, the earlier work in probability modeling seems to have been ignored by some more contemporary workers, without reasons being indicated.

The results of G/NG studies are clearly of great interest to food producers and food safety managers. It is perhaps time, then, that the G/NG modeling community seeks to find common ground and to begin to develop a rigorous framework for the development, and interpretation, of growth limits studies.

APPENDIX A3.1 — CHARACTERIZATION OF ENVIRONMENTAL PARAMETERS AFFECTING MICROBIAL KINETICS IN FOODS

In most situations, temperature is the major environmental parameter in uencing kinetics of microorganisms in food and its effect is included in most predictive microbiology models. During processing, storage, and distribution the temperature of foods can vary substantially, frequently including periods of temperature abuse for chilled foods (see, e.g., Audits International, 1999; James and Evans, 1990; O'Brien, 1997; Sergelidis et al., 1997). Thus, it is an important property of secondary models that they can predict the effect of changing temperatures on microbial kinetics and application of these models relies on information about product temperature and its possible variation over time. Numerous types of thermometers, temperature probes, and data loggers are available (McMeekin et al., 1993, pp. 257-269; seagrant.oregonstate.edu/extension/ sheng/loggers.html) to measure the temperature of foods or food processing equipment. Infrared non-contact thermometers are often appropriate for foods but their use is limited for process equipment with stainless surfaces.

Foods are typically stored aerobically, vacuum packed, or by using modi ed atmosphere packing (MAP). "Controlled atmosphere packaging" can be considered a special case of MAP. MAP foods are exposed to an atmosphere different from both air and vacuum packed usually involving mixtures of the gasses carbon dioxide (CO2), nitrogen (N2), and oxygen (O2).

O2 and CO2 in uence growth of most microorganisms and secondary predictive models must take their effect into account. The solubility of O2 in water, and thereby into the water phase of foods, is low (~0.03 l/l) but it can be important for growth and metabolism of microorganisms in both aerobic and MAP-stored products (Dainty and Mackey, 1992). Numerous techniques and instruments are available to determine O2 in the gas phase or dissolved in food. Microelectrodes to determine gradients of dissolved O2 in foods are available (www.instechlabs.com/oxy-gen.html; www.microelectrodes.com/) but models to predict the effect of such gradients remain to be developed. To account for the effect of aerobic or vacuum packed storage of foods a categorical approach has been used within predictive microbiology. For aerobic conditions growth media with access to air have been agitated. For vacuum packed foods microorganisms typically have been grown under 100% N2.

CO2 inhibits growth of some microorganisms substantially and, to predict microbial growth in MAP foods, it is important to determine the equilibrium concentration in the gas phase or the concentration of CO2 dissolved into the foods water phase. At equilibrium, the concentration of CO2 dissolved into the water phase of foods is proportional to the partial pressure of CO2 in the atmosphere surrounding the product. Henry's law (Equation A3.1) provides a good approximation for the solubility of CO2.

In Equation A3.1, KH is Henry's constant (mg/l/atm) and pCO2 is the partial pressure (atm) of CO2. Between 0 and 160°C the temperature dependence of the Henry's constant can be predicted by Equation A3.2:

exp(-6.8346 +1.2817 ■ 104 / K - 3.7668 ■ 106 / K2 + 2.997 ■ 103 / K3)

where K is the absolute temperature (Carroll et al., 1991). Those authors expressed Kh as MPa/mole fraction. In Equation A3.2 the constants 101,325 Pa/atm and 2.4429 was used to convert this unit into mg CO2/l H2O/atm.

For MAP foods in exible packaging the partial pressure of CO2 is conveniently determined from the percentage of CO2 inside the pack. A range of analytical methods is available to determine CO2 concentration in gas mixtures or concentrations of dissolved CO2 (Dixon and Kell, 1989; www.pbi-dansensor.com/Food.htm).

As shown from Equation A3.1 and Equation A3.2, the concentration of CO2 dissolved in the water phase of a MAP food with 50% CO2 in the headspace gas at equilibrium is 1.67 g/l at 0°C and 1.26 g/l at 8°C. Because of the high solubility of

CO2 in water the gas composition in the headspace of MAP foods changes after packaging. The equilibrium gas composition is in uenced by several factors, e.g., the percentage of CO2 in the initial headspace gas (%CO2Imtial), the initial gas/product volume ratio (G/P), temperature, pH, lipids in the food, respiration of the food, and, of course, permeability of the packing lm. Different mass-balance equations to predict the rate of adsorption and solubility of CO2 have been suggested (Devlieghere et al., 1998; Dixon and Kell, 1989; Gill, 1988; Lowenadler and Ronner, 1994; Simpson et al., 2001a,b; Zhao et al., 1995). In chilled foods the rate of absorption of CO2 is rapid compared to growth of microorganisms. Therefore, to predict micro-bial growth in these MAP foods it is suf cient to take into account the equilibrium concentration of CO2.

Devlieghere et al. (1998) suggested Equation A3.3 to predict the concentration of CO2 in the water phase as a function of °%CO2Imtial and G/P. In Equation A3.3, dCO2 is the density of CO2 (1.976 g/l).

G ■ dCO2 + Kh 1 - (— ■ Kh ■G ■ %COinitial ■ dCO2

Equation A3.3 does not take into account the effect of the storage temperature and Devlieghere et al. (1998) developed a polynomial model to predict the concentration of dissolved CO2 as a function of %CO2Initial, G/P, and temperature. If, for example, %CO2Initial is 25, the polynomial model predicts that a G/P ratio of three results in higher concentration of dissolved CO2 than does a G/P ratio of 4, which is not logical. In contrast we have found that the combined use of Equation A3.2 and Equation A3.3 provides realistic predictions for concentrations of dissolved CO2. It also seems relevant to include the effect of product pH on dissolved CO2, and thereby the equilibrium concentration of CO2 in the gas phase of MAP foods.

A3.1.3 Salt, Water-Phase Salt, and Water Activity

While temperature is the single most important storage condition in uencing growth of microorganisms in foods, NaCl is the most important product characteristic in many foods. The concentration of NaCl in foods can be determined as chloride by titration (Anon., 1995a). Instruments to determine NaCl indirectly from conductivity measurements are available but extensive calibration for particular types of products may be required. In fresh and intermediate moisture foods, NaCl is dissolved in the water phase of the products.

To predict the effect of NaCl on growth of microorganisms in these products the concentration of water-phase salt (WPS) or relative humidity, i.e., the water activity (aw) must be determined (Equation A3.4 to Equation A3.7).

Water-phase salt can be calculated from Equation A3.4:

% Water phase salt = %NaCl (w/v) x 100/(100 - % dry matter +%NaCl (w/v)) (A3.4)

Water activity is a fundamental property of aqueous solutions and is de ned as:

^ Po where p is the vapor pressure of the solution and p0 is the vapor pressure of the pure water under the same conditions of temperature, etc.

For mixtures of NaCl and water there is a direct relation between the WPS content and aw (Resnik and Chirife, 1988; Equation A3.6 and Equation A3.7). For cured foods where NaCl is the only major humectant these relations are valid as documented, e.g., for cold-smoked salmon (Jergensen et al., 2000) and processed "delicatessen" meats (Ross and Shadbolt, 2001). To determine water activity of foods, instruments relying on the dew point method are now widely used because of their speed (providing results within a few minutes), robustness, and reliability but other methods and instruments are available (Mathlouthi, 2001).

aw = 1 - 0.0052471 • %WPS- 0.00012206 • %WPS2 (A3.6)

% WPS = 8 -140.07 • (aw - 0.95) - 405.12 • (aw - 0.95)2 (A3.7)

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